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Prog. Theor. Phys. Vol. 43 No. 3 (1970) pp. 808-829

[ Full Text PDF : FREE ACCESS (1040K) ]

Some Mathematical Aspects of Multiple Poles in the Off-Shell Scattering Amplitude

Masakuni Ida

Institute of Physics, College of General Education, University of Tokyo, Meguro-ku, Tokyo

(Received October 4, 1969)

Abstract:

An attempt at a general theory of the Bethe-Salpeter equation in a previous work is continued, with our attention focussed on the problem of multiple poles in the off-shell scattering amplitude. Basic assumptions made in the previous work reduce the problem to that of multiple poles in the resolvent of a compact operator, which is hermitian analytic as an operator-valued function of P4. Properties of compact operators are discussed on the basis of the Riesz-Schauder theory. Multiple poles are characteristic of non-normal operators, which are shown to be considerably different from hermitian operators in many points of physical importance. Several quantities useful for the understanding of multiple poles are introduced that their properties are investigated. Multiple poles in the resolvent of an analytic operator are studied as a limit of degeneracy of several simple poles.


URL : http://ptp.ipap.jp/link?PTP/43/808/
DOI : 10.1143/PTP.43.808

[ Full Text PDF : FREE ACCESS (1040K) ] Citation:

References:

  1. M. Ida, Prog. Theor. Phys. 43 (1970), 184[PTP].
  2. W. Heisenberg, Nucl. Phys. 4 (1957), 532[CrossRef].
  3. N. Nakanishi, Phys. Rev. 140 (1965), B947[APS]; ibid. 147 (1966), 1153[APS].
  4. K. Seto, Prog. Theor. Phys. 41 (1969), 1340[PTP].
  5. N. Nakanishi, Prog. Theor. Phys. 39 (1968), 1585[PTP]. See also reference 8).
  6. S. Naito, Prog. Theor. Phys. 41 (1969), 500[PTP].
  7. J. Arafune, Prog. Theor. Phys. 40 (1968), 620[PTP].
  8. N. Nakanishi, Prog. Theor. Phys. 41 (1969), 233[PTP].
  9. a) F. Riesz and B. Sz-Nagy, Functional Analysis (Frederick Ungar Pub. Co., New York, 1965).
    b) A. C. Zaanen, Linear Analysis (North-Holland Pub. Co., Amsterdam, 1964).
    c) N. Dunford and J. T. Schwartz, Linear Operators (Interscience Publishers, Inc., New York, 1964), Part I and II.
  10. E. R. Lorch, Trans. Am. Math. Soc. 45 (1939), 217.
    See also Examples 21, 22 and 23 in Chap. 7 of reference 9b).

Citing Article(s) :

  1. Progress of Theoretical Physics Vol. 44 No. 5 (1970) pp. 1317-1339 :
    The Reggeization of the Bethe-Salpeter Scattering Amplitudes
    Hideki Matsumoto
  2. Progress of Theoretical Physics Vol. 79 No. 3 (1988) pp. 627-644 :
    An Effective Hamiltonian in Nonequilibrium Thermo Field Dynamics with a Reservoir
    Hideki Matsumoto
  3. Progress of Theoretical Physics Vol. 107 No. 4 (2002) pp. 679-688 :
    Quantum Phase Coordinate as a Zero-Mode in Bose-Einstein Condensed States
    Hideki Matsumoto and Shoichi Sakamoto

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